Technical companion: the common chronological construction

1. Row changes and ordered measurement

For ordinary compatible rows, $R_d(b,r,L)=(b+d,r-d,L)$ and $A_e(b,r,L)=(b,r+e,L+e)$ . Any compatible row difference has the unique parameters $d=\Delta b,e=\Delta L$ . These algebraic operations commute on unrestricted triples. When nonnegative ages or source-selected states are required, each intermediate row must also be admissible.

The nineteen matched MT/LXX rows have the following nonzero supports:

Source change	Rows	Amplitude
Begetting change	Adam, Seth, Enosh, Kenan, Mahalalel, Enoch; Arphaxad, Shelah, Eber, Peleg, Reu, Serug	$u=100$
Begetting change	Nahor	$u/2=50$
Lifespan change	Lamech	$\lambda=-24$
Lifespan change	Arphaxad, Shelah	$a=27$
Lifespan change	Eber	$e=40$
Lifespan change	Peleg, Reu, Serug	$u=100$
Lifespan change	Nahor	$n=60$

The 38-output change matrix has rank five. Imposing the fitted relation $e+n=u$ reduces the parameter count to four; neither description claims to generate the baseline source rows. The matched head measurements are

\Delta_{\rm Reg}=12.5u,\qquad \Delta_{\rm Cum}=3u+\lambda+2a+e+n.

At the source values these are 1250 and 430. Native Cainan contributes a labelled insertion with measures 130 and 460.

Let $U$ be the suffix-sum matrix for a fixed ordered path. With common terminal, $X'-X=U(w'-w)$ ; with changed terminal, add its difference to each boundary. Adjacent differences recover the weights. Thus the complete field retains location, while its head alone cannot recover the distribution.

Source/proof route: complete source-row packet; whole-genealogy five-amplitude matrix; C1162–C1189; Regular staircase C1253–C1264.